Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
Newton's Method is an iterative numerical technique used to approximate the roots of a real-valued function. It involves using the derivative of the function to iteratively converge on a root starting from an initial guess.
5 Must Know Facts For Your Next Test
Newton’s Method requires an initial guess, and its accuracy depends heavily on this choice.
The formula for Newton's Method is $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$.
Convergence is typically quadratic if the method converges, meaning errors decrease exponentially with each iteration.
If the derivative $f'(x)$ is zero or close to zero, Newton’s Method may fail or provide inaccurate results.
Newton's Method can diverge if the initial guess is not sufficiently close to the actual root.
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Related terms
Derivative: A measure of how a function changes as its input changes; it represents the slope of the function at any point.
Root-finding Algorithm: A computational algorithm used to find solutions where a given function equals zero.
Quadratic Convergence: A property of an iterative method where errors decrease proportionally to the square of previous errors in successive iterations.